Stable equilibria of the angular collision operator in higher dimensions

Establish the existence of stable equilibria for the angular collision operator C(f) = ∇u·(∇u f − (λ/Du) f ∇u(χ^2 u^T (ρf Qf) u)) on the unit sphere S^{n−1} in dimensions n ≥ 4, which is required to extend the hydrodynamic limit beyond n ∈ {2,3}.

Background

The hydrodynamic limit in the paper relies on the characterization of stable equilibria of the operator C, following Degond–Frouvelle–Liu (KRM 2022). This characterization is currently available for n ∈ {2,3}.

Extending the macroscopic derivation to higher dimensions requires the existence of stable equilibria for C in n ≥ 4, which remains an unresolved issue. Proving such existence would likely involve a careful analysis of the variational structure and spectral properties of C in higher dimensions.